Quantum conductance

1
Quantum conductance

• I.A. Shelykh
• St. Petersburg State Polytechnical University,
  St. Petersburg, Russia
• International Center for Condensed Matter
  Physics, Brasilia, Brazil

ICCMP
2
Outline

• Overwiew of the classical results
• Quantum Point Contacts
• 1D Ballistic Conductance and Landauer Buttiker
  formula
• Quantum interference and Aharonov-Bohm effect
• Integer and Fractional Quantum Hall effect
• 0.7 anomaly and fractional quantization of
  ballistic conductance

3
Classical results
Ohm Law
Consequent
G1
G2
G-1G1-1G2-1
W
Parallel
G1
L
G2
GG1G2
4
Quantum Point Contacts
Let us consider a very small object (QPC or
QWire) LltltLfree, WkF-1
The condition LltltLfree means that there is no
inelastic scattering within the region of the QPC
Is G8 then?
5
Contact resistance
The condition WkF-1 means that in the region of
QPC the motion in x-direction is quantized
The origin of the resistance redistribution of
the current among the current-carrrrying modes at
the interfaces
µ
E0
For parabolic confinement
Right lead
Left lead
QPC
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Ballistic conductance
T0
R. Landauer. IBM J. Res. Dev., 1, 233 (1957)
I
µ1
µ2
Vds0
Vdsgt0
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Ballistic conductance staircase
B. J. van Wees, Phys. Rev. Lett. 60, 848-850
(1988) D. A. Wharam et al, J. Phys. C 21
L209-L214 (1988)
If there are N open subbands
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The role of backscattering
Gi
Gc
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Several scatterers
?
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Effects of quantum interference
LltltLf
Quantum interference term
Fabry-Perot oscillations of quantum conductance
N.T. Bagraev et al, Semiconductors, 34, 817
(2000)
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Parallel connection
No interference NN1N2, GG1G2
With interference
To account for the round trips scattering matrix
S
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Aharonov-Bohm effect
How one can easily change phaseshift between the
electrons propagating in the quantum
ring? Possible way apply a magnetic flux through
the ring
F
Strong backscattering AAS half-period
oscillations
Weak backscattering AB oscillations
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Classical Hall effect
y
UH
x
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Experimental configuration
L
V1
V2
I
W
V3
?xy
?xx
B
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Landau quantization
Group velocity
DOS
16
Edge states
y
x
vg?0
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Ballistic conductance and QHE
I
?µeVH
I-
µ
Backscattering is supressed
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Quantum Hall Effect (QHE)
K. v. Klitzing, G. Dorda, and M. Pepper Phys.
Rev. Lett. 45, 494-497 (1980)
Classical result
In the experiment
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Fractional QHE
D.C. Tsui et al, PRL 48, 1559 (1982)
H.L. Stormer et al, PRL 50, 1953 (1982)
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Interpretation of FQHE
Laughlin wavefunction
Composite fermions
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Fractional quantization of the ballistic
conductance ( 0.7 anomaly 
K.J. Thomas et al, PRL 77, 135 (1996)
Related with spin!
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Singlet and Triplet Scattering
V.V. Flambaum, M.Yu. Kuchiev, PRB 62, R7869
(2000) T. Rejec et al, J. Phys. Cond. Matt. 12,
L233 (2000)
Localised and propagating electrons interact in
the region of the QPC
Eigenstates singlet and triplet configurations.
The probabilities of realization
23
Singlet and Triplet Scattering
V.V. Flambaum, M.Yu. Kuchiev, PRB 62, R7869
(2000) T. Rejec et al, J. Phys. Cond. Matt. 12,
L233 (2000)
Localised and propagating electrons interact in
the region of the QPC
Eigenstates singlet and triplet configurations.
The probabilities of realization
24
Singlet and Triplet Scattering
V.V. Flambaum, M.Yu. Kuchiev, PRB 62, R7869
(2000) T. Rejec et al, J. Phys. Cond. Matt. 12,
L233 (2000)
Localised and propagating electrons interact in
the region of the QPC
Eigenstates singlet and triplet configurations.
The probabilities of realization
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0.75 structure calculation
Consider the case
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Is fractional ballistic conductance universal?
D.J. Reilly et al, PRB 63, R121311 (2001)
?
For short constriction
For long wire
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QPC with Large Spin
I.A. Shelykh et al, PRB 74, 085322 (2005)
Supposing the contact containing a total spin J
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Fractional quantization calculation
The Hamiltonian
Using the following basis
One represents H in a block-diagonal form
Diagonalised Hamiltonian reads

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With increase of the length of the wire J
increases and conductance decreases- as in
experiment!
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Spontaneous polarization of 1D electron gas
2 for unpolarized 1 for polarized
Chuan-Kui Wang, K.-F. Berggren PRB 57, 4552
(1998) N.T. Bagraev et al PRB 70, 155315 (2004)
Why big J can appear in long quantum wires? Due
to exchange interaction!
Calculation gives
Qualitatively in 1D
Critical density
Dominant for high density
Dominant for low density
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What happens with holes? Light and Heavy Hole
Bands in a QPC
Si / GaAs / Ge
Bands splitted in energy depending on the width
of the QPC
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Spin Dependent Scattering for Holes
Initial state
Conductance at T 0 (44 transmission
amplitudes)
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Model
Matrix form (16x16)
where
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Physical Origin of the Plateaus
States presenting total spin ST 3 7 states
ST 1 3 states ST 2 5 states ST
0 1 states.
Potential Barriers
Ferromagnetic Interaction
Steps at
Antiferromagnetic Interaction
Steps at
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Si
Ferromagnetic
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Si
Antiferromagnetic
Antiferromagnetic
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Applying an Axial Magnetic Field
Si
Ferromagnetic
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Experiment for the holes
L.P. Rokhinson et al, 2006
N.T. Bagraev et al 2002
Klochan et al, 2006
????
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Thank you for your attention Obrigado por a sua
atenção ??????? ?? ????????